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Re: Well, we should check out/up on every possibility.

Well, we should check out (= make a systematic inspection of; examine the validity of;verify; examine or investigate) every possibility.

Well, we should check (= find out;make an inquiry into, search through; investigate; compare and examine to prove true or correct;investigate or verify;inspect so as to determine accuracy, quality, or other condition; test) every possibility.

Well, we should check up (see below) every possibility.

Well, we should check up on ( = test the truth or soundness of; find out about; seek more info; investigate, scrutinize, or inspect; compare and examine to prove true or correct) every possibility.

As can be seen, the above 4 sentences “make sense and carry about the same idea.”